A cochain-level Adem relation?

The following article might be relevant. This is the review I wrote for it a number of years ago.

MR1622630 (99h:55027) Reviewed Real, Pedro(E-SEVLIS-AM1) On the computability of the Steenrod squares. (English, Italian summary) Ann. Univ. Ferrara Sez. VII (N.S.) 42 (1996), 57–63 (1998). 55S05 (55S10)

Let $C_∗(X)$ denote the normalized chains on a simplicial set $X$. The Eilenberg-Zilber theorem can be proved using very explicit operators $AW:C_∗(X×Y) \rightarrow C_∗(X)\otimes C_∗(Y)$, $EM:C_∗(X)\otimes C_∗(Y) \rightarrow C_∗(X×Y)$, and $SHI:C_∗(X×Y)→C_{∗+1}(X×Y)$. The contracting homotopy SHI is least familiar, but like EM it involves an explicit summation over shuffles. The author gives an explicit and clean cochain level formula for the mod 2 Steenrod operation $Sq^i$ in terms of AW and SHI. Like SHI, it is exponential in terms of computational complexity. {Reviewer's remark: The author systematically uses the word "idempotent'' where he seems to mean "involution''.}

Explicit homotopies for a cochain-level Adem relation were first worked out in:

Greg Brumfiel, Anibal M. Medina-Mardones, John Morgan. A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra. 2020

The odd-prime case is still open.